Investigation of Receptive Fields Using Representations of the Dihedral Groups

Abstract We investigate families of receptive fields (i.e., low-level filter systems) that receive their inputs from sensors located on a finite, regular grid. We will first introduce a class of models that describe the behavior of such systems. We introduce the representation theory of the dihedral groups to derive some important properties of such systems that originate in the structure of the grid (and not in the particular nature of the system). We will show that representation theory leads directly to algorithms with a structure similar to those of the FFT. We demonstrate possible applications of the theory in the field of low-level vision by showing how to construct and analyze different types of filter families. All these different approaches are simplified by the same, universal, coordinate transformation. In the second part of the paper we demonstrate how the proposed method can be used in the computation of the Karhunen-Loeve expansion. We show that the correlation matrix of an arbitrary stochastic process that has the symmetry of the dihedral group is block diagonalized by the coordinate transformation developed in the first part of the paper. In our experiments we investigate images from four different data sets: one set consisting of 39 images of size 512 2 , one set consisting of 55 images of size 256 2 , one set consisting of 1000 of size 256 2 captured from a satellite TV channel, and finally 2000 images of size 256 2 from a video tape. We show that in all cases the eigenvectors computed from the block-diagonal approximation of the correlation matrix are good approximations of the eigenvectors computed from the full correlation matrix. We show with an image coding example that both sets of eigenvectors (the vectors computed from the full matrix and the vectors computed from the approximation) give practically indistinguishable results in this application.